The private sector is incapable of delivering complete Health Care for Everyone.

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Cons

The private sector is optimising for profits, not for people's healthcare.

No system is capable of delivering complete "Health Care for Everyone". Therefore, it follows that the "private sector" is incapable.

A primarily for-profit system does not have an incentive to deliver healthcare to as many consumers as possible. It has an incentive to deliver healthcare to as many consumers as it can while remaining profitable. By definition, this excludes those whose healthcare is less likely to yield profit - the chronically ill, old, infirm, etc.

Lifetime caps on healthcare spending, a well-known feature of the private sector healthcare system, explicitly sets a profitability number to a human life. If you cost more than $X to insure, you lose us too much money, and you're done being covered.

The private sector naturally seeks to innovate and improve health care in order to be chosen above competition while remaining accessible to all as to not miss out on any potential patients.